Type-II topoisomerase (type-II topo) activity is essential for maintaining the topological state of genomes and hence the survival of all living systems. This fact is underscored by the existence of at least one type-II topoisomerase gene in all known organisms. Targeting of type-II topo activity is the basis for many anticancer and anti-microbial drugs, making this strategy one of the most successful general therapeutic approaches in the last thirty years. However, many detailed aspects of type-II topo mechanisms remain unknown or poorly characterized and better insight into aspects of topoisomerase enzymology will be needed for future drug development as increasing resistance to existing topoisomerase inhibitors evolves. The objective of the present project is to develop a comprehensive mathematical model of topological states in knotted and/or supercoiled DNA that explains important details of type-II-topoisomerase mechanisms. We will test predictions of these models by using state-of-the-art single-molecule fluorescence techniques. These are challenging, but achievable, goals, the outcome of which will improve our basic understanding of DNA enzymology and anti-topoisomerase drug activity. This proposal builds on fundamental progress that we have made in understanding DNA-loop mechanics and DNA topology to develop a theoretical description of the structure of complex nucleoprotein assemblies. Our approach involves a fusion of mathematical knot theory with semi-analytical and numerical models of the statistical thermodynamics of DNA conformations. We propose to study the action of type-II enzymes on circular DNA in terms of transitions between topological states defined by knot type and linking number in knotted, supercoiled DNA. The resulting probability distributions of topological states strongly depend on rates of type-II enzyme-induced transitions, and thus provide a probe of enzymatic mechanisms of type-II topoisomerases.